Rose-Hulman Undergraduate Mathematics Journal
V OLUME ∞, I SSUE π, 2112

A fetching and descriptive title for yourmanuscript
By Euclid of Alexandria*, Leonard Euler†, and Johann Peter Gustav Lejeune Dirichlet‡

Abstract. Insert your abstract here. The abstract should be a short paragraph that details what thecurrent paper sets out to do, and a high-level description of how it goes about what it does. Also, you
should avoid putting citations, like this one [3], in the abstract.

1 Introduction
Ideally, the introduction should introduce the background of the problem at hand andmotivate why mathematicians (and/or others) are interested in studying it. This should
be aimed at the level of undergraduates who are not experts in your area. We’d also likethe introduction to contain a road map for the article; i.e., a description of what to expectin each section.

2 The title of the second section
The class file contains definitions for several environments (e.g., thm, corollary,lemma, etc). Examples follow below.
Important definitions should be set in the defin environment.
Definition 2.1. Let p ∈ N and suppose the following hold:

P1. If d ∈ N such that d | p, then d = 1 or d = p.
P2. p 6= 1.
Then we say that p is a prime number.
* Thanks to all those hapless Pythagoreans.
†‡Thanks for nothing, Bernoulli.
You should really be thanking me.
Mathematics Subject Classification. 11A41
Keywords. primes

1

Short title

2

Use the label command for reference to these environments later on in your document. The class file uses the cleveref package which determines the type of label beingreferenced:
LATEX code
Using ref:Using cref:

Result\textbf{definition \ref{prim-def}} definition 2.1\cref{prim-def}
definition 2.1

Lemma 2.2. If n ∈ N, then there exists a prime number p such that p | n.
Proof. Suppose, for contradiction, that there exists n ∈ N such that n > 1 and n is notdivisible by any prime. LetS = {n ∈ N : n > 1, n is not divisible by a prime}.
Then S is a non-empty subset of N. The well-ordering principle1 fashions a least elementof S, say m. Note that m is not prime (otherwise it would be divisible by a prime, namelyitself ). Since m is not prime, there exists a ∈ N such that a | m and a 6= 1 and a 6= m.
Let b ∈ Z such that ab = m. Note that b ∈ N since a, m ∈ N. What’s more, b > 1 sinceotherwise a = m. So it must be that 1 < a, b < m. Since divisibility is transitive, it follows
that neither a nor b is divisible by a prime. But then a ∈ S (and so is b) and a = m/b < mcontradicting the minimality of m. Thus it must be that S is empty which proves thelemma.
Theorem 2.3 (Euclid). There are infinitely many prime numbers.
Proof. Let p 1 , p 2 , . . . , p k be primes, and let
n = p 1 p 2 · · · p k + 1.
By lemma 2.2, there exists a prime p such that p | n. Suppose, for contradiction, thatp = p i for some i = 1, 2, . . . , k. It follows that p | n − p 1 p 2 · · · p k = 1, but this is absurd. So it
must be that p 6= p i for any i = 1, 2, . . . k. This shows that a new prime can be constructedfrom any finite list of primes. Hence the number of primes is not finite.
Use the cite command to cite references. For example, the above proof can be foundin Euclid’s Elements [1]. See below for how to add citations to the thebibliographyenvironment.
1

An excellent principle.

Rose-Hulman Undergrad. Math. J.

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Euclid, Euler, and Dirichlet

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3 A title for the third section
There is an example environment.
Example 3.1. The geometric series∞ 1X
k=0 2kconverges to 2.
When starting a sentence with a reference, use the Cref command. Example 3.1 canbe used to prove that 2 is not the only prime.
Theorem 3.2. The set of primes includes more numbers than just 2.
Proof. If 2 were the only prime number, then the fundamental theorem of arithmetic
would give us that∞ 1∞ 1
XX=.
kn=1 nk=0 2
The series on the left diverges whereas the series on the right converges. This is impossible, so there must be more prime numbers beyond the number 2.
There is a remark environment as well.
Remark 3.3. The idea in theorem 3.2 can be extended to show that the set of primes
includes more numbers than just 2 and 3. In fact, Euler went on to prove that if thenumber of primes were finite, then the harmonic series would converge.

4 A title for the fourth section
There are prop and corollary environments as well.
Proposition 4.1. Every natural number can be written uniquely as a product of a squarefree number and a square.
As a corollary to proposition 4.1 we obtain
Corollary 4.2. Let π(n) denote the number of primes less than or equal to n. Thenπ(n) ≥log n.
2 log 2Proof. There are no more than 2π(n) square free numbers less than n. Also, there are nop
more than n squares less than n. It follows from proposition 4.1 thatpn ≤ 2π(n) n.
The corollary follows by applying log.
Rose-Hulman Undergrad. Math. J.

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Short title

4For an unnumbered remark, use the xrem environment.
Remark. Note that corollary 4.2 implies that there are infinitely many primes.

References[1] Euclid, Euclid’s Elements, the Thomas L. Heath translation, Green Lion Press, SantaFe, NM, 2002. MR1932864[2] Euler, Foundations of differential calculus, translated from the Latin by John D.
Blanton, Springer-Verlag, New York, 2000. MR1753095[3] P. G. L. Dirichlet, Gedächtnißrede auf Carl Gustav Jacob Jacobi, in Nachrufe auf
Berliner Mathematiker des 19. Jahrhunderts, 6–34, Teubner-Arch. Math., 10, Teubner, Leipzig. MR1104895
Euclid of Alexandria
Ancient Greece

u-kid@ancient-greece.biz
Leonard Euler
University of Baseloiler@notmail.com
Johann Peter Gustav Lejeune Dirichlet
University of Bonndirichleet@yahoo.net

Rose-Hulman Undergrad. Math. J.

Volume ∞, Issue π, 2112